Yes, it will then be a tetrahedron.
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The base is different. A rect. pyramid has a rectangle as a base. Because the sides are different length for a rectangle, the triangles will not all be congruent (equal). A square pyramid has a square as the base. All sides of the square are the same length, so all the triangular sides of the pyramid will be congruent.
A square pyramid is a pyramid with a square base. It has 4 outer faces, all of which are triangles, that meet at the same point perpendicular to the centre of the base. All pyramids have at least 4 faces (including the base), where the outer faces are all triangular while the base can be trilateral, quadrilateral, or any other polygon. The quadrilateral form (the square pyramid) is the most common form of pyramid, and is the form associated with the Egyptian pyramids.
Any triangle-based pyramid will do. If the sides are all the same length it's then called a tetrahedron.
In general, an octagonal pyramid is a pyramid with an octagon for a base and triangles (which all have a common vertex) for sides. The octagon my or may not be a regular one.
Imagine a pyramid, with a triangle shaped base, with three sides coming up each of which is a triangle. That is a tetrahedron (4 sides, all the same side, with each one the shape of a triangle). * * * * * Mostly correct, but the four faces need not all be the same.